Problem: Find the value of $x$ for which the matrix
\[\begin{pmatrix} 1 + x & 7 \\ 3 - x & 8 \end{pmatrix}\]is not invertible.
Solution: A matrix is not invertible if and only its determinant is 0.  This gives us the equation
\[(1 + x)(8) - (7)(3 - x) = 0.\]Solving, we find $x = \boxed{\frac{13}{15}}.$